Elizabeth's reaction re: Ric's fingerprints on the gun makes sense. She doesn't love Ric, but he's demonstrated that he's very determined to get a "second chance" with her and she's afraid of what could happen if she and Nik became romantically involved again. Plus she figures Lucky would hate them both forever long-distance.
I think the tears came from frustration and embarrassment that she thinks she's just continued to believe in men who lie to her/hurt people - Lucky, Australian doc, AJ, and now Ric again. (I don't include Jason because I remember she called him an "honest man.) I understand that she would like to believe she could be happy with someone other than Nikolas, but Ric is the kind of guy who should never get a chance with her again ... if for no other reason than a needy, obsessive guy who uses her son for manipulative purposes is a BAD example as a male role model for boys.
Ric is now in the role that AJ had in the love triangle with Liz and Nik, with a variation of the downfall storyline. However, Liz had genuinely developed feelings for AJ before Nik returned to Port Charles,AJ was clearly not ready to be in a healthy adult relationship due to all his issues, and he and Elizabeth did not have a history. With AJ, there was a give-and-take, she liked spending time with him, and wanted to see where they could go. With Ric, she all but said she was settling.